PROBABILITY WORD PROBLEMS ON DICE AND COINS

Problem 1 :

Two unbiased dice are rolled once. Find the probability of getting

(i) a doublet (equal numbers on both dice)

(ii) the product as a prime number

(iii) the sum as a prime number

(iv) the sum as 1

Solution :

Sample space  =  {(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)(2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) }

n(S)  =  36

(i) a doublet (equal numbers on both dice)

Let "A" be the event of getting doublet

A = {(1, 1) (2, 2) (3, 3) (4, 4) (5, 5) (6, 6)}

n(A)  =  6

P(A)  =  n(A) / n(S)

 P(A)  =  6/36  =  1/6

(ii) the product as a prime number

Let "B" be the event of the product as a prime number

B = {(1, 2) (1, 3) (1, 5) (2, 1) (3, 1) (5, 1)}

n(B)  =  6

P(B)  =  n(B)/n(S)

P(B)  =  6/36  =  1/6

(iii) the sum as a prime number

Let "C" be the event of getting the sum as a prime number

C = {(1, 1) (1, 2) (1, 4) (1, 6) (2, 1) (2, 3) (2, 5) (3, 2) (3, 4) (4, 1) (4, 3) (5, 2) (5, 6) (6, 1) (6, 5) }

n(C)  =  15

P(C)  =  n(C) / n(S) 

P(C)  =  15/36

P(C)  =  5/12

(iv) the sum as 1

Let "D' be the event of getting the sum as 1.

Because it is impossible event, P(D)  =  0

Problem 2 :

Three fair coins are tossed together. Find the probability of getting

(i) all heads

(ii) atleast one tail

(iii) atmost one head

(iv) atmost two tails

Solution :

Sample space  =  {HHH. HHT, HTH, HTT, THH, THT, TTH, TTT}

n(S)  =  8

(i) all heads

Let "A" be the event of getting all heads 

A = {HHH}

n(A)  =  1

  p(A)  =  n(A)/n(S)  

  P(A)  =  1/8

(ii) atleast one tail

Let "B" be the event of getting atleast one tail

B = {HHT, HTH, HTT, THH, THT, TTH, TTT}

n(B)  =  7

P(B)  =  n(B)/n(S)

P(B)  =  7/8

(iii) atmost one head

Let "C" be the event of getting at most one head

C = {THT, TTH, TTT}

n(C)  =  3

P(C)  =  n(C)/n(S)  

P(C)  =  3/8

(iv) atmost two tails

Let "D" be the event of getting atmost two tails

D = {HHH. HHT, HTH, HTT, THH, THT, TTH}

n(D)  =  7

P(D)  =  n(D)/n(S)

P(D)  =  7/8

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